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Question
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
`"y"^2 = "a"("b - x")("b + x")`
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Solution
`"y"^2 = "a"("b - x")("b + x") = "a"("b"^2 - "x"^2)`
Differentiating both sides w.r.t. x, we get
`"2y" "dy"/"dx" = "a" (0 - 2"x") = - 2 "ax"`
∴ `"y" "dy"/"dx" = - "ax"` ....(1)
Differentiating again w.r.t. x, we get
`"y" * "d"/"dx" ("dy"/"dx")^2+ "dy"/"dx" * "dy"/"dx" = - "a" xx 1`
∴ `"y" ("d"^2"y")/"dx"^2 + ("dy"/"dx")^2 = - "a"`
∴ `"xy" ("d"^2"y")/"dx"^2 + "x" "dy"/"dx" = - "ax"`
∴ `"xy" ("d"^2"y")/"dx"^2 + "x" ("dy"/"dx")^2 = "y" "dy"/"dx"` ....[By (1)]
∴ `"xy" ("d"^2"y")/"dx"^2 + "x" "dy"/"dx" - "y" "dy"/"dx" = 0`
This is the required D.E.
Notes
The answer in the textbook is incorrect.
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